A closed box with a square base is to have a volume of 2000 cm3. The material for the top and bottom of the box costs P30 per square centimeter and the material for the sides cost P15 per square centimeter. Find the dimensions of the box so that the total cost of material is the least possible and all its dimensions do not exceed 20 cm.
Minimization
In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.
Maxima and Minima
The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.
Derivatives
A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.
Concavity
In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.
A closed box with a square base is to have a volume of 2000 cm3. The material for the top and bottom of the box costs P30 per square centimeter and the material for the sides cost P15 per square centimeter. Find the dimensions of the box so that the total cost of material is the least possible and all its dimensions do not exceed 20 cm.
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